Source: Adv. in Appl. Probab.
Volume 44, Number 2
Hazard rates play an important role in various areas, e.g. reliability theory,
survival analysis, biostatistics, queueing theory, and actuarial studies.
Mixtures of distributions are also of great preeminence in such areas as most
populations of components are indeed heterogeneous. In this study we present a
sufficient condition for mixtures of two elements of the same natural
exponential family (NEF) to have an increasing hazard rate. We then apply this
condition to some classical NEFs having either quadratic or cubic variance
functions (VFs) and others as well. Particular attention is paid to the
hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF,
and the NEF generated by Kummer distributions of type 2. The application of
such a sufficient condition is quite intricate and cumbersome, in particular
when applied to the latter three NEFs. Various lemmas and propositions are
needed to verify this condition for such NEFs. It should be pointed out,
however, that our results are mainly applied to a mixture of two populations.
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